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Variationofactivationvolumewith
temperatureforFe,Si,andGe
ARTICLEinMATERIALSLETTERS·SEPTEMBER2003
ImpactFactor:2.49·DOI:10.1016/S0167-577X(03)00310-0
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Materials Letters 57 (2003) 4319 – 4322
www.elsevier.com/locate/matlet
Variation of activation volume with temperature for Fe, Si, and Ge
K.K. Mani Pandey, Om Prakash1, B. Bhattacharya *
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
Received 27 January 2003; received in revised form 4 April 2003; accepted 7 April 2003
Abstract
In this paper, a new approach is presented to calculate the activation volume, which is based on Eyring absolute reaction rate
theory. Emphasis is placed on the determination of activation volume from the indentation creep microhardness data measured
using Vickers indentor at constant load for various dwell times and temperatures. Different materials like Fe (with metallic
bonding), Si, and Ge (with covalent bonding) are chosen for the study. The results serve to validate the approach outlined here
because direct comparison can be made with the data obtained through a conventional creep test of specimens. The result
obtained also shows that activation volume increases with increasing homologous temperature.
D 2003 Elsevier Science B.V. All rights reserved.
Keywords: Activation volume; Indentation; Vickers; Microhardness
1. Introduction
The deformation of crystalline materials at elevated
temperatures is a thermally activated process. A
crystal contains a distribution of obstacles of activation energy ( Q) towards the motion of dislocation
during the deformation. Each obstacle is characterized
by critical stress r0, which means that at 0 K, these
obstacles are overcome by the dislocation under an
applied stress r = r0. If a constant stress r is applied to
the crystal, obstacles with r0 < r will be overcome at
once by the dislocation, giving rise to instantaneous
strain. When the measuring temperature is greater
* Corresponding author. GE India Technology Center, Department of Mechanical Engineering, Whitefield, Bangalore 560066,
India. Tel.: +91-512-2597824 (Off.), +91-512-2597913 (Lab.);
fax: +91-512-2597408, +91-512-2590007.
E-mail address: bishakh@iitk.ac.in (B. Bhattacharya).
1
Present address: GE India Technology Center, Whitefield,
Bangalore 560066, India.
than 0 K, obstacles with r0 > r may be overcome
due to thermal activation with a probability proportional to exp Q=ðkB T Þ . Hence, at higher temperatures, as soon as the dislocation is arrested at an
obstacle with r0 slightly higher than r, the thermal
energy predominates the small value of Q. This enables the dislocation to overcome the obstacle, resulting
in rapid strain.
2. Model
Under the thermally activated process, the dislocation can jump over the barrier with frequency m+ [1],
which is given by Eyring equation:
Q þ Va r
þ
m ¼ m0 exp
ð1Þ
kB T
where m0 is the attempt frequency, Va is the activation
volume, kB is the Boltzmann constant, and T is
0167-577X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0167-577X(03)00310-0
4320
K.K. Mani Pandey et al. / Materials Letters 57 (2003) 4319–4322
In an indentation creep test, the experimental data
are in the form of microhardness variation with dwell
time, where hardness itself is calculated from the
applied load and the indentation size. For Vickers
indenter with an included angle of 136j, microhardness
(H) is given by [4]:
H ¼ 2 sin 68j
P
MPa
D2
ð6Þ
where P is applied load [in N] and D is length of the
indentation diagonal [in mm]. The strain rate was
defined as [5]:
Fig. 1. Variation of lnðð1=HÞðdH=dtÞÞ with H/(kBT ) for Fe.
ės ~
temperature [in K]. Once the barrier has been overcome and the dislocation segment has fallen into the
next trough, it acquires new energy to overcome the
following barrier in a purely random process. Thus,
there will be a chance to jump back with frequency:
m ¼ m0 exp
Q Va r
:
kB T
1 dD
D dt
using Eqs. (6) and (7), the strain rate may be written
as:
ės ~ 1 1 dH
2 H dt
ð2Þ
ės ¼ K1
Thus, the net forward reaction rate is given by:
m ¼ mþ m
Va r
Q
m ¼ m0 sinh
exp
:
kB T
kB T
ð7Þ
ð3Þ
1 dH
H dt
ð8Þ
where K1 is a proportionality constant. Following
Roebuck and Alomand [5] and Evans and Goetze
[10], stress r is related to microhardness by the relation
The net forward reaction determines the macroscopically observed creep strain rate of specimen
subjected to external loading at a specified temperature; hence, steady strain rate ės is given by [2,3]:
ės ¼ AT
Va r
Q
exp
kB T
kB T
ð4Þ
where A is a constant. For moderate stress and narrow
temperature range, Eq. (4) is simplified to the exponential form as [4]:
Q þ Va r
ės ¼ Aexp
:
kB T
ð5Þ
Fig. 2. Variation of lnðð1=HÞðdH=dtÞÞ with H/(kBT) for Si.
K.K. Mani Pandey et al. / Materials Letters 57 (2003) 4319–4322
4321
Table 2
Activation volumes of test materials at different temperatures
Fig. 3. Variation of lnðð1=HÞðdH=dtÞÞ with H/(kBT) for Ge.
r = H/3. We substitute Eq. (5) into Eq. (8) and rearrange:
1 dH
Va H
ð9Þ
ln þB¼
H dt
3kB T
where B is defined as:
B ¼ ln K1 ln A þ
Q
:
kB T
ð10Þ
The plots of lnðð1=HÞðdH=dt ÞÞ versus H/(kBT)
represent a straight line with intercept B and slope Va/3,
where slopes directly measure the activation volume.
3. Results and discussion
Indentation creep response is analyzed using the
data of iron [6], silicon, and germanium [7]. The
variations of lnðð1=HÞðdH=dt ÞÞ versus H/(kBT) are
shown in Figs. 1– 3 for Fe, Si, and Ge, respectively.
The test materials, their melting point, Burger vector,
Table 1
Test materials, their melting points, Burger vectors (b), crystal
structures, and their bonding
SN Test
Melting Burger
Crystal
materials point (K) vector (A) structure
1
2
3
Fe
Si
Ge
1530
1687
1211
2.48
3.83
3.99
Type of
bonding
bcc
Metallic
fcc (diamond type) Covalent
fcc (diamond type) Covalent
SN
Test
materials
Test
temperature (K)
T/Tm
Activation
volume (b3)
1
2
3
4
5
6
7
8
Fe
Fe
Fe
Si
Si
Ge
Ge
Ge
644
755
866
1173
1273
873
1073
1173
0.42
0.49
0.56
0.70
0.75
0.72
0.88
0.96
37.8
66.6
115.2
23.4
25.4
9.0
14.4
18.0
crystal structure, and nature of bonding are given in
Table 1 [8], while the activation volumes of the test
materials determined from the graph are given in
Table 2 in terms of b3 (where b is the Burger vector).
The results indicate that activation volume generally increases with increasing test temperature. For
iron, the activation volume is 37.8b3, 66.6b3, and
115.2b3 at a homologous temperature of 0.42, 0.49,
and 0.56 K, respectively, which is consistent with the
earlier observed value [9]. The activation volume for
silicon is 23.4b3 and 25.4b3 at a homologous temperature of 0.70 and 0.75 K. The activation volume for
germanium 9.0b3, 14.4b3, and 18.0b3 at 0.72, 0.88,
and 0.96 K, respectively.
4. Conclusion
The temperature and time dependence of Vickers
microhardness data of test materials are the result of
plastic flow by glide process. This is a consequence of
thermally activated motion of dislocations. Silicon
and germanium have an ‘fcc’ structure and each atom
has four nearest neighbours to which it is linked by
four purely covalent bonds. On the other hand, iron
possesses a ‘bcc’ structure at the temperature under
consideration and has metallic bond. The iron, silicon,
and germanium microhardness data are measured at
the same stress. It is observed that the activation
volume of silicon (23.4b3, 25.4b3) and germanium
(9.0b3, 14.4b3, and 18.0b3) even at extremely high
homologous temperatures is lower than that of iron
(37.8b3, 66.6b3, and 115.2b3). This is due to the
highly directional covalent bond of silicon and germanium in comparison to the weak metallic bond of
iron.
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K.K. Mani Pandey et al. / Materials Letters 57 (2003) 4319–4322
Acknowledgements
The authors are grateful to the Council of Scientific
and Industrial Research (CSIR), New Delhi, for the
financial assistance provided to this research.
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